Explicit formula for N_n(F) over finite fields

Develop an explicit formula for the number N_n(F) of isomorphism classes (under T ~ S induced by A(T) ≅ A(S)) when F is a finite field, as a function of n and the size of F.

Background

After introducing invariants (the Wall and the Measure sequence) and proving the lower bound N_n ≥ C_{n−1}, the authors note that exact enumeration remains difficult, particularly for larger n because related systems of equations become complicated.

They ask specifically for a closed-form expression for N_n(F) when F is finite, which would complement their lower bounds and the finiteness results in this case.